**32n – 1 is divisible by 8, for all natural numbers .**

###### Answers:

Let P(n): 3 – 1 is divisible by 8, for all natural numbers n.

Now, P(l): 3 – 1 = 8, which is divisible by 8.

Hence, P(l) is true.

Let us assume that, P(n) is true for some natural number n = k.

P(k): 3^{2k} – 1 is divisible by 8

or 32k -1 = 8m, m ∈ N (i)

Now, we have to prove that P(k + 1) is true.

P(k+ 1): 3^{2}(k+1)– l

= 3^{2k }• 3^{2} — 1

= 9(8m + 1) – 1 (using (i))

= 72m + 9 – 1

= 72m + 8

= 8(9m +1), which is divisible by 8 Thus P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for all natural numbers n.